Effective Index¶

Effective Index of the Fundamental Mode¶

Effective index of the fundamental mode of a waveguide can be accessed by using neff function of effective_index module. Note that for the default parameters below, this mode happens to be a TE mode. The function takes the following arguments:

• wavelength: Wavelength of the light in the waveguide, in micrometers. A list of wavelengths can also be passed. Wavelength range is from 1.2 μm to 1.7 μm; it will return 0 for wavelengths out of range.
• width: Width of the waveguide, in micrometers. A list of widths can also be passed. Width range is from 0.240 μm to 0.700 μm; it will return 0 for widths out of range.

In [1]:

from sipkit.effective_index import neff
import jax.numpy as jnp

neff(width=0.5, wavelength=1.55)

from sipkit.effective_index import neff import jax.numpy as jnp neff(width=0.5, wavelength=1.55)

WARNING:jax._src.lib.xla_bridge:No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)

Out[1]:

DeviceArray(2.44523381, dtype=float64)

Use Cases¶

• Scalar width, scalar wavelength

In [2]:

neff(width=0.5, wavelength=1.55)

neff(width=0.5, wavelength=1.55)

Out[2]:

DeviceArray(2.44523381, dtype=float64)

• Scalar width, 1D Array wavelength

In [3]:

neff(width=0.5, wavelength=jnp.linspace(1.4, 1.5, 5))

neff(width=0.5, wavelength=jnp.linspace(1.4, 1.5, 5))

Out[3]:

DeviceArray([2.6138077 , 2.58583721, 2.55782222, 2.52975709, 2.50163816],            dtype=float64)

• 1D Array width, scalar wavelength

In [4]:

neff(width=jnp.linspace(0.4, 0.5, 5), wavelength=1.55)

neff(width=jnp.linspace(0.4, 0.5, 5), wavelength=1.55)

Out[4]:

DeviceArray([2.22706193, 2.29512748, 2.35308258, 2.40263554, 2.44523381],            dtype=float64)

• 1D Array width, 1D Array wavelength

In [5]:

neff(width=jnp.linspace(0.4, 0.5, 3), wavelength=jnp.linspace(1.4, 1.5, 5))

neff(width=jnp.linspace(0.4, 0.5, 3), wavelength=jnp.linspace(1.4, 1.5, 5))

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
/tmp/ipykernel_264/3451282617.py in <module>
----> 1 neff(width=jnp.linspace(0.4, 0.5, 3), wavelength=jnp.linspace(1.4, 1.5, 5))

    [... skipping hidden 11 frame]

~/checkouts/readthedocs.org/user_builds/sipkit/envs/1.4.0/lib/python3.7/site-packages/sipkit/effective_index.py in neff(width, wavelength)
     69             (width - width_min) * ((width_size - 1) / (width_max - width_min)),
     70         ],
---> 71         order=1,
     72     )
     73 

~/checkouts/readthedocs.org/user_builds/sipkit/envs/1.4.0/lib/python3.7/site-packages/jax/_src/scipy/ndimage.py in map_coordinates(input, coordinates, order, mode, cval)
    139     input: ArrayLike, coordinates: Sequence[ArrayLike], order: int, mode: str = 'constant', cval: ArrayLike = 0.0,
    140 ):
--> 141   return _map_coordinates(input, coordinates, order, mode, cval)

    [... skipping hidden 5 frame]

~/checkouts/readthedocs.org/user_builds/sipkit/envs/1.4.0/lib/python3.7/site-packages/jax/_src/scipy/ndimage.py in _map_coordinates(input, coordinates, order, mode, cval)
    119       contribution = input_arr[indices]
    120     else:
--> 121       all_valid = functools.reduce(operator.and_, validities)
    122       contribution = jnp.where(all_valid, input_arr[indices], cval)
    123     outputs.append(_nonempty_prod(weights) * contribution)

    [... skipping hidden 1 frame]

~/checkouts/readthedocs.org/user_builds/sipkit/envs/1.4.0/lib/python3.7/site-packages/jax/_src/numpy/lax_numpy.py in deferring_binary_op(self, other)
   4934     args = (other, self) if swap else (self, other)
   4935     if isinstance(other, _accepted_binop_types):
-> 4936       return binary_op(*args)
   4937     if isinstance(other, _rejected_binop_types):
   4938       raise TypeError(f"unsupported operand type(s) for {opchar}: "

    [... skipping hidden 5 frame]

~/checkouts/readthedocs.org/user_builds/sipkit/envs/1.4.0/lib/python3.7/site-packages/jax/_src/numpy/ufuncs.py in <lambda>(x1, x2)
     81     fn = lambda x1, x2: lax_fn(*_promote_args_numeric(numpy_fn.__name__, x1, x2))
     82   else:
---> 83     fn = lambda x1, x2: lax_fn(*_promote_args(numpy_fn.__name__, x1, x2))
     84   fn.__qualname__ = f"jax.numpy.{numpy_fn.__name__}"
     85   fn = jit(fn, inline=True)

    [... skipping hidden 7 frame]

~/checkouts/readthedocs.org/user_builds/sipkit/envs/1.4.0/lib/python3.7/site-packages/jax/_src/lax/lax.py in broadcasting_shape_rule(name, *avals)
   1571         result_shape.append(non_1s[0])
   1572       else:
-> 1573         raise TypeError(f'{name} got incompatible shapes for broadcasting: '
   1574                         f'{", ".join(map(str, map(tuple, shapes)))}.')
   1575 

TypeError: and got incompatible shapes for broadcasting: (5,), (3,).

• Scalar width, 2D Array wavelength

In [6]:

neff(width=0.45, wavelength=jnp.array([[1.4, 1.5], [1.6, 1.7]]))

neff(width=0.45, wavelength=jnp.array([[1.4, 1.5], [1.6, 1.7]]))

Out[6]:

DeviceArray([[2.53952222, 2.41551475],
             [2.2904706 , 2.16536619]], dtype=float64)

• 2D Array width, scalar wavelength

In [7]:

neff(width=jnp.array([[0.4, 0.45], [0.5, 0.55]]), wavelength=1.55)

neff(width=jnp.array([[0.4, 0.45], [0.5, 0.55]]), wavelength=1.55)

Out[7]:

DeviceArray([[2.22706193, 2.35308258],
             [2.44523381, 2.51410111]], dtype=float64)

• 2D Array width, 2D Array wavelength

In [8]:

waveguide_width = jnp.array([0.5, 0.6])
wavelength = np.array([1.5, 1.6, 1.7])
waveguide_width, wavelength = np.meshgrid(waveguide_width, wavelength)
neff(waveguide_width, wavelength)

waveguide_width = jnp.array([0.5, 0.6]) wavelength = np.array([1.5, 1.6, 1.7]) waveguide_width, wavelength = np.meshgrid(waveguide_width, wavelength) neff(waveguide_width, wavelength)

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
/tmp/ipykernel_264/1394243315.py in <module>
      1 waveguide_width = jnp.array([0.5, 0.6])
----> 2 wavelength = np.array([1.5, 1.6, 1.7])
      3 waveguide_width, wavelength = np.meshgrid(waveguide_width, wavelength)
      4 neff(waveguide_width, wavelength)

NameError: name 'np' is not defined

Effective Index of Higher Order Modes¶

Effective Index of first 5 modes in a waveguide can be accessed by using the following functions of effective_index module:

• neff_te0: Effective index of TE0 mode. (Note that currently this is the same as neff, due to the waveguide width boundaries used.)
• neff_tm0: Effective index of TM0 mode.
• neff_te1: Effective index of TE1 mode.
• neff_tm1: Effective index of TM0 mode.
• neff_te2: Effective index of TE3 mode.

In [9]:

from sipkit.effective_index import neff_te0, neff_tm0


neff_te0_list = neff_te0(width=jnp.linspace(0.4, 0.5, 5), wavelength=1.55)
print(neff_te0_list)

neff_tm0_list = neff_tm0(width=jnp.linspace(0.4, 0.5, 5), wavelength=1.55)
print(neff_tm0_list)

from sipkit.effective_index import neff_te0, neff_tm0 neff_te0_list = neff_te0(width=jnp.linspace(0.4, 0.5, 5), wavelength=1.55) print(neff_te0_list) neff_tm0_list = neff_tm0(width=jnp.linspace(0.4, 0.5, 5), wavelength=1.55) print(neff_tm0_list)

[2.22706193 2.29512748 2.35308258 2.40263554 2.44523381]
[1.69157614 1.71344278 1.73396505 1.75313724 1.77099004]

For the cases where the effective index of a mode does not exist, the function will return index of oxide layer as the effective index at that wavelength.

In [10]:

from sipkit.effective_index import neff_te2


neff_te2_list = neff_te2(width=jnp.linspace(0.4, 0.5, 5), wavelength=1.55)
print(neff_te2_list)

from sipkit.effective_index import neff_te2 neff_te2_list = neff_te2(width=jnp.linspace(0.4, 0.5, 5), wavelength=1.55) print(neff_te2_list)

[1.4440023 1.4440023 1.4440023 1.4440023 1.4440023]

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Last update: 2022-12-28
Created: 2022-10-19